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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 66270.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66270.k1 | 66270m7 | \([1, 0, 1, -11781748, 15564503978]\) | \(16778985534208729/81000\) | \(873116441649000\) | \([2]\) | \(2543616\) | \(2.4892\) | |
66270.k2 | 66270m8 | \([1, 0, 1, -1001828, 52446506]\) | \(10316097499609/5859375000\) | \(63159464818359375000\) | \([2]\) | \(2543616\) | \(2.4892\) | |
66270.k3 | 66270m6 | \([1, 0, 1, -736748, 242879978]\) | \(4102915888729/9000000\) | \(97012937961000000\) | \([2, 2]\) | \(1271808\) | \(2.1427\) | |
66270.k4 | 66270m5 | \([1, 0, 1, -637343, -195893692]\) | \(2656166199049/33750\) | \(363798517353750\) | \([2]\) | \(847872\) | \(1.9399\) | |
66270.k5 | 66270m4 | \([1, 0, 1, -151363, 19510316]\) | \(35578826569/5314410\) | \(57285169736590890\) | \([2]\) | \(847872\) | \(1.9399\) | |
66270.k6 | 66270m2 | \([1, 0, 1, -40913, -2888944]\) | \(702595369/72900\) | \(785804797484100\) | \([2, 2]\) | \(423936\) | \(1.5934\) | |
66270.k7 | 66270m3 | \([1, 0, 1, -29868, 6499306]\) | \(-273359449/1536000\) | \(-16556874745344000\) | \([2]\) | \(635904\) | \(1.7961\) | |
66270.k8 | 66270m1 | \([1, 0, 1, 3267, -220472]\) | \(357911/2160\) | \(-23283105110640\) | \([2]\) | \(211968\) | \(1.2468\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 66270.k have rank \(0\).
Complex multiplication
The elliptic curves in class 66270.k do not have complex multiplication.Modular form 66270.2.a.k
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.